As noted by @Kusalananda in the comments (and this mostly unrelated Stack Overflow answer), `Ctrl`+``` typically returns an ASCII `NUL`/`0` in most terminals.
`Ctrl`+`Space` and (very oddly) `Ctrl`+`2` do the same.
@Kusalananda also mentioned a Vim trick that appears to also work in Tmux:
set-option -g prefix C-@
Which I've confirmed works. Note that, as a result, `Ctrl`+`Space` and `Ctrl`+`2` will then also work as prefix, assuming that nothing else in the OS or terminal captures them instead.